Gaussian Process (GP) regression is a non-parametric Bayesian machine learning method [1]} that aims to infer an unknown function \(f:\mathbb {R}^n\rightarrow \mathbb {R}\)  from noisy data.
A standard assumption of GP regression is that \(f\)  is a sample from a GP with zero mean and covariance \(\kappa : \mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathbb {R}\) .
Let \(\mathrm {D}=\lbrace (\mathrm {x}_i,\mathrm {y}_i),i\in \lbrace 1,\dots ,m \rbrace \rbrace \)  be a dataset, where
\(\mathrm {y}_i\)  is a sample of an observation of \(f(\mathrm {x}_i)\)  with independent zero-mean noise \({v}\) , which is assumed to be normally distributed with variance \(\sigma ^2\)  and \(\mathrm {X}\)  and \(\mathrm {Y}\)  be ordered vectors with all points in \(\mathrm {D}\)  such that \(\mathrm {X}_i = \mathrm {x}_i\)  and \(\mathrm {Y}_i = \mathrm {y}_i\) . Further, call \(K(\mathrm {X},\mathrm {X})\)  the matrix with \(K_{i,j}(\mathrm {X}_i,\mathrm {X}_j)=\kappa (\mathrm {x}_i,\mathrm {x}_j)\) , \(K(\mathrm {x},\mathrm {X})\)  the vector such that \(K_{i}(\mathrm {x},\mathrm {X})=\kappa (\mathrm {x},\mathrm {X}_i)\) , and \(K(\mathrm {X},\mathrm {x})\)  defined accordingly. Then, the predictive distribution of \(f\)  at a test point \(\mathrm {x}\)  is given by the conditional distribution of \(f\) 
given \(\mathrm {D}\) , which is Gaussian with mean \(\mu _\mathrm {D}\)  and variance \(\sigma _\mathrm {D}^2\)  given by
\(\begin{split}& \mu _{\mathrm {D}}(\mathrm {x}) = K(\mathrm {x},\mathrm {X}) \big ( K(\mathrm {X},\mathrm {X})+ \sigma ^2I_{m} \big )^{-1} Y \\\end{split}\\\begin{split}& \sigma _{\mathrm {D}}^2(\mathrm {x}) = \kappa (\mathrm {x},\mathrm {x})- K(\mathrm {x},\mathrm {X})\big ( K(\mathrm {X},\mathrm {X})+ \sigma ^2 I_{m} \big )^{-1}K(\mathrm {X},\mathrm {x}),\end{split}\) 
